# Two-Point Local Optimality in k-Means via Boundary-Point Screening

arXiv: https://arxiv.org/abs/2610.06182 · HTML: https://mlsanity.com/paper/2610.06182/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 28 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Builds on (most cited first, 8 of 8)

- [Mind the duality gap: safer rules for the Lasso](https://mlsanity.com/paper/1505.03410/) (2015)
- [Fast K-Means with Accurate Bounds](https://mlsanity.com/paper/1602.02514/) (2016)
- [From TOWER to SPIRE: Adding the Speech Modality to a Translation-Specialist LLM](https://mlsanity.com/paper/2503.10620/) (2025)
- [Multi-Swap k-Means++](https://mlsanity.com/paper/2309.16384/) (2023)
- [Towards Calibrated Deep Clustering Network](https://mlsanity.com/paper/2403.02998/) (2024)
- [Superpixel Graph Contrastive Clustering with Semantic-Invariant Augmentations for Hyperspectral Images](https://mlsanity.com/paper/2403.01799/) (2024)
- [Modified K-means Algorithm with Local Optimality Guarantees](https://mlsanity.com/paper/2506.06990/) (2025)
- [Further heuristics for k-means: The merge-and-split heuristic and the (k,l)-means](https://mlsanity.com/paper/1406.6314/) (2014)

## Abstract (arXiv)

Lloyd's algorithm and the discrete local (D-local) optimization method (Li et al., 2025) for k-means provide only weak local-optimality guarantees, and their solution quality remains sensitive to initialization. In this paper, we introduce r-point local optimality, under which no reassignment of at most r samples decreases the objective function, and focus on r=2. The main computational obstacle is the O(n2(k2+d)) cost of exhaustive two-point certification for n samples in d dimensions and k clusters. To address this challenge, we prove that (i) every improving two-point move of a D-local optimum must involve a cluster shared by both reassignments, and (ii) only certificate-defined boundary points can participate in an improving pair. Exploiting this structure, we propose Boundary-Point-Screened Two-Point Local Search (BPS-2PLS), which terminates at a two-point local optimum. For fixed k,d and nonvanishing cluster occupancy, the number m of retained candidates satisfies m=O_mathbbP(log n) under i.i.d. sampling from a bounded-support distribution with bounded density or from a Gaussian mixture. Across twelve benchmarks, BPS-2PLS attains the lowest available mean WCSS on ten. In a subsampling study, screening retains 0.10% to 2.81% of samples on average at the largest tested sizes. The code is available at https://github.com/lwl-learning/BPS-2PLS.